nLab integer Heisenberg group

1. Idea

The integer Heisenberg groups or discrete Heisenberg groups are non-abelian discrete central extensions of the integer product groups $\mathbb{Z}^{2g}$, $g \in \mathbb{N}$, analogous to how the ordinary Heisenberg groups are non-abelian central extensions of $\mathbb{R}^{2g}$. Generally there are analogous $R$-Heisenberg groups for any commutative ring $R$.

In fact, for even multiples of the basic such extension, integer Heisenberg groups are discrete subgroups of ordinary $\mathbb{R}$-Heisenberg groups covering a lattice inclusion $\mathbb{Z}^{2g} \hookrightarrow \mathbb{R}^{2g}$ (cf. Prop. 2.4 below).

In physics, specifically in quantum field theory, these even-weight integer Heisenberg groups are closely related to quantum abelian Chern-Simons theory (spelled out below, following SS25, see also at 2-Cohomotopy moduli of surfaces).

2. Definition

Integer Heisenberg group extensions of $\mathbb{Z}^2$

We discuss the restriction of the basic $\mathbb{R}$-Heisenberg group $H_3$ (see there) along the inclusion $\mathbb{Z}^2 \hookrightarrow \mathbb{R}^2$ and the mod-$n$ quotient groups of that.

In the following, we denote cyclic groups, for $n \in \mathbb{N}$, uniformly by

$$\mathbb{Z}_{k} \;\coloneqq\; \left\{ \begin{array}{lcl} \mathbb{Z} &\vert& k = 0 \\ \mathbb{Z}/(k) &\vert& \text{otherwise} \mathrlap{\,.} \end{array} \right.$$

Recall (see there) the equivalence of central extensions of a discrete group $G$ by an abelian group $A$ (for the present case equipped with the trivial $G$-action) with the degree=2 group cohomology $H^2(G;A)$.

Proposition 2.1. For $k \in \mathbb{N}$, the central group extensions of $\mathbb{Z}^2$ by $\mathbb{Z}_k$ are all integral multiples of the following elementary Heisenberg extension:

(1)$$\left\{ \begin{array}{l} \big(a,b,[n]\big) \,\in\, \mathbb{Z} \times \mathbb{Z} \times \mathbb{Z}_{k} \,, \\ \big( a,b,[n] \big) \cdot \big( a',b',[n'] \big) \;=\; \big( a + a', b + b', [ n + n' + {\color{red} a \, b' } ] \big) \end{array} \right\} \,.$$

Proof. The central extensions in question

$$1 \to \mathbb{Z}_k \xrightarrow{cntrl} H \longrightarrow \mathbb{Z}^2 \to 1$$

are classified by the degree=2 group cohomology of $\mathbb{Z}^2$ with coefficients in $\mathbb{Z}_k$ (equipped with the trivial $\mathbb{Z}^2$-action).

That second group cohomology is:

(2)$$\begin{array}{ccl} H^2_{Grp}(\mathbb{Z}^2;\, \mathbb{Z}_k) &\simeq& H^2(B \mathbb{Z}^2;\, \mathbb{Z}_k) \\ &\simeq& H^2(T^2;\, \mathbb{Z}_k) \\ &\simeq& H^2(S^1_a \vee S^1_b \vee S^2;\, \mathbb{Z}_k) \\ &\simeq& H^2(S^2;\, \mathbb{Z}_k) \\ &\simeq& \mathbb{Z}_k \mathrlap{\,,} \end{array}$$

where we used, in order of the appearance:

1. that group cohomology is ordinary cohomology of the domain group’s classifying space;

2. that the classifying space of the integers is the circle and that this respects Cartesian products;

3. that the stable homotopy type of the 2-torus is the wedge sum of two circles with a 2-sphere (see there).

The generator of this cohomology group must be the group 2-cocycle

$$\alpha \,\in\, H^2_{Grp}\big( \mathbb{Z}^2 ;\, \mathbb{Z}_k \big)$$

given by

(3)$$\begin{array}{ccc} \mathbb{Z}^2 \times \mathbb{Z}^2 &\xrightarrow{\; \alpha \;}& \mathbb{Z}_{n} \\ \big( (a,b), (a',b') \big) &\mapsto& [ a b'] \mathrlap{\,,} \end{array}$$

whose cocycle condition $[a b'] + [(a+a')b''] = [a(b'+b'')] + [a' b'']$ is evidently satisfied due to the distributive law in the ring $\mathbb{Z}$ and which is clearly not divisible.  ▮

Example 2.2. In the integer Heisenberg group (1)

• the inverse elements are given by

  $$\big(a ,\, b ,\, [n]\big)^{-1} \;=\; \big( -a ,\, -b ,\, [-n + a b] \big)$$

• the basic group commutators give

  (4)$$\begin{array}{ccl} \big[ (1,0,0), (0,1,0) \big] &\equiv& (1,0,0) \cdot (0,1,0) \cdot (1,0,0)^{-1} \cdot (0,1,0)^{-1} \\ &=& (1,1,1) \cdot (-1,0,0) \cdot (0,-1,0) \\ &=& (0,1,1) \cdot (0,-1,0) \\ &=& (0,0,1) \mathrlap{\,.} \end{array}$$

Remark 2.3. The basic integer Heisenberg group extension (1), at $k = 0$, is isomorphic to the matrix group of $3 \times 3$ upper triangular matrices with integer entries and all “1”s on their diagonal, since matrix multiplication gives:

$$\left[ \begin{matrix} 1 & a & c \\ 0 & 1 & b \\ 0 & 0 & 1 \end{matrix} \right] \cdot \left[ \begin{matrix} 1 & a' & c' \\ 0 & 1 & b' \\ 0 & 0 & 1 \end{matrix} \right] \;=\; \left[ \begin{matrix} 1 & a + a' & c + c' + a b' \\ 0 & 1 & b + b' \\ 0 & 0 & 1 \end{matrix} \right]$$

In this matrix form the basic integer Heisenberg group $H_3(\mathbb{Z})$ is commonly known in the pure group theory-literature (e.g. Dummit & Foote 2006 p 35, Budylin 2014) where the higher level extensions typically do not find attention.

But the relation to the real Heisenberg group (in both senses of “real”) is made by the second multiple of this basic extension (cf. Gelca & Uribe 2010 p. 7, Gelca & Hamilton 2012 Def 2.4, Gelca & Hamilton 2015 Def 2.3):

Proposition 2.4. Twice the basic integer Heisenberg extension (3) is isomorphic to

(5)$$\left\{ \begin{array}{l} \big(a, b, [n]\big) \,\in\, \mathbb{Z} \times \mathbb{Z} \times \mathbb{Z}_{k} \,, \\ \big( a, b, [n] \big) \cdot \big( a', b', [n'] \big) \;=\; \big( a + a' ,\, b + b' ,\, [n + n' + {\color{red} a \cdot b' - a' \cdot b}] \big) \end{array} \right\} \mathrlap{\,.}$$

Proof. It is sufficient to see that the second red summand in (5) is a group 2-cocycle cohomologous to the first red summand, which means equivalently that the 2-cocycle

$$\big( (a, b), (a', b') \big) \;\mapsto\; a \cdot b' + a' \cdot b$$

has a coboundary: This coboundary is readily found to be

$$(a, b) \;\mapsto\; - a \cdot b \,,$$

since

$$- a \cdot b - a' \cdot b' \,+\, (a + a') \cdot (b + b') \;=\; a \cdot b' + a' \cdot b \,.$$

  ▮

3. Properties

Basic properties

Proposition 3.1. The multiple=2 discrete Heisenberg group $\mathbb{Z}_{k} \to \widehat{\mathbb{Z}^{2}} \to \mathbb{Z}^2$ (5) is the following quotient group of the product group $F(a,b) \times \mathbb{Z}_{k}$ with the free group $F(a,b)$ on two generators:

(6)$$\widehat{\mathbb{Z}^2} \,\simeq\, \big( F(a,b) \times \mathbb{Z}_{k} \big) \big/ \big( a \cdot b = [2] \cdot b \cdot a \big) \,.$$

Proof. The map

$$\begin{array}{ccc} \frac{ F(a,b) \times \mathbb{Z}_{k} }{ a \cdot b = [2] \cdot b \cdot a } &\xrightarrow{\;}& \widehat{\mathbb{Z}^2} \\ [a] &\mapsto& (1,0,0) \\ [b] &\mapsto& (0,1,0) \\ [n] &\mapsto& (0,0,[n]) \end{array}$$

is clearly a bijection on underlying sets, and is a group homomorphism since the quotient relation (6) is respected in $\widehat{\mathbb{Z}^2}$, by (4).  ▮

(cf. also arXiv:2203.08030, p 21)

Linear representations

Example 3.2. For $k \in \mathbb{Z}$, linear representations of twice the $\mathbb{Z}_{2k}$-Heisenberg extension of $\mathbb{Z}^2$ (5)

(7)$$\begin{array}{ccc} \widehat{\mathbb{Z}^2} & \xrightarrow{\; W \;} & \mathrm{Aut}\big( \mathscr{H}_1 \big) \end{array}$$

are given by:

$$\mathscr{H}_1 \;\coloneqq\; Span_{\mathbb{C}}\Big( {\vert 0 \rangle} ,\, {\vert 1 \rangle} ,\, \cdots ,\, {\vert k-1 \rangle}, \Big)$$

$$\begin{array}{ccccccl} W_a & \coloneqq & W(1,0,0) &\colon& {\big\vert n \big\rangle} &\mapsto& \zeta^{2n} {\big\vert n \big\rangle} \\ W_b &\coloneqq& W(0,1,0) &\colon& {\big\vert n \big\rangle} &\mapsto& {\big\vert (n + 1) \,mod\, k \big\rangle} \\ \zeta &\coloneqq& W(0,0,1) &\colon& {\big\vert n \big\rangle} &\mapsto& \zeta {\big\vert n \big\rangle} \mathrlap{\,,} \end{array}$$

for $\zeta$ a $2k$th root of unity.

For instance:

(8)$$W_a \cdot W_b \cdot \zeta^{-1} \;=\; W(1,1,0) \;=\; W_b \cdot W_a \cdot \zeta^{+1} \,.$$

Notation: In the following we write “$[n]$” for “$n \,mod\, k$” and “$\sum_{[n]}$” for “$\sum_{n =0}^{k-1}$”.

Proof. To see that this is a linear representation, by (6) it is sufficient to check that the basic group commutator is represented, in that

(9)$$W_a \cdot W_b \,=\, \zeta^2 \, W_b \cdot W_a \mathrlap{\,,}$$

which is evidently the case, since

$$\begin{array}{ccl} W_a \cdot W_b {\big\vert [n] \big\rangle} &=& W_a \cdot {\big\vert [n+1] \big\rangle} \\ &=& \zeta^{2(n+1)} \, {\big\vert [n+1] \big\rangle} \mathrlap{\,,} \end{array}$$

while

$$\begin{array}{ccl} W_b \cdot W_a {\big\vert [n] \big\rangle} &=& \zeta^{2n} \, W_b {\big\vert [n] \big\rangle} \\ &=& \zeta^{2n} \, {\big\vert [n + 1] \big\rangle} \mathrlap{\,.} \end{array}$$

  ▮

For more on linear representations of the level$=1$ integer Heisenberg group, see Floratos & Tsohantjis 2022.

Remark 3.3. (relation to quantum states and quantum observables of abelian Chern-Simons theory)
The complex dimension of the representation $\mathscr{H}_1$ (7) is that expected for the space of quantum states of abelian Chern-Simons theory on a 2-torus (cf. Manoliu 1998a p 40, Gelca & Uribe 2010 Prop. 2.2)

(10)$$dim(\mathscr{H}_1) \;=\; K \,,$$

and the group commutator-relation (9)

$$W_a \cdot W_b = \zeta^2 \, W_b \cdot W_a$$

of linear operators acting on this irrep (7) reflects the characteristic commutator-relation of Wilson loop-quantum observables in abelian Chern-Simons theory on a 2-torus (cf. Tong 2016 (5.28), p 166).

Modular automorphisms

Moreover, the integer Heisenberg group at leve 2 knows about the modular group acting on its irrep (7), hence on the space of quantum states $\mathscr{H}_{T^2}$ from Rem. 3.3, hence about the “modular functor” of abelian Chern-Simons theory:

Definition 3.4. (modular action on the integer Heisenberg group)
Since the colored summand in (5) is the canonical symplectic form on $\mathbb{Z}^2$, the integer symplectic group in dimension 2, hence the modular group

$$\Big\{ g \in \mathrm{GL}_2(\mathbb{Z}) \,\Big\vert\, \underset{ \mathrm{det}(g) \cdot (a b' - a' b) }{ \underbrace{ (g_{1 1} a + g_{12} b) (g_{2 1} a' + g_{22} b') - (g_{11} a' + g_{12}b') (g_{21} a + g_{22} b) } } = a b' - a'b \Big\} \;\simeq\; \mathrm{SL}_2(\mathbb{Z}) \,,$$

acts by evident group automorphism on (5):

(11)$$\begin{array}{ccc} \mathrm{SL}_2(\mathbb{Z}) \times \widehat{\mathbb{Z}^2} &\xrightarrow{}& \widehat{\mathbb{Z}^2} \\ \Big( g, \big(a,b,[n]\big) \Big) &\mapsto& \Big( g_{11}a + g_{12}b ,\, g_{21}a + g_{22}b ,\, [n] \Big) \mathrlap{\,.} \end{array}$$

Remark 3.5. Recall (from there) that the modular group $SL_2(\mathbb{Z})$ is the subgroup of the general linear group generated by the two elements

(12)$$S \;\coloneqq\; \left[ \begin{array}{rr} 0 & 1 \\ -1 & 0 \end{array} \right] \;\;\;\;\; \text{and} \;\;\;\;\; T \;\coloneqq\; \left[ \begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array} \right] \mathrlap{\,.}$$

and presented via these generators subject to the following relations:

(13)$$SL_2(\mathbb{Z}) \;\simeq\; \big\langle S, T \,\big\vert\, S^4 = \mathrm{e} ,\, (T S)^3 = \mathrm{e} ,\, S^2 (T S) = (T S)S^2 \big\rangle \,.$$

Proposition 3.6. For even $k$ (10) there exists a linear representation

(14)$$\begin{array}{ccc} \mathrm{SL}_2(\mathbb{Z}) \times \mathscr{H}_1 &\xrightarrow{\;}& \mathscr{H}_1 \mathrlap{\,,} \end{array}$$

of the modular group on the underlying complex vector space of (7), which intertwines the action (7) of $\widehat{\mathbb{Z}^2}$ on $\mathscr{H}_1$, with its automorphic images under the modular group action (11) on the Heisenberg group, in that:

(15)$$m(W) \cdot m\Big( {\big\vert [n] \big\rangle} \Big) \;=\; m\Big( W \cdot {\big\vert [n] \big\rangle} \Big) \,, \;\;\;\;\; \forall \left\{ \begin{array}{ccc} m &\in& SL_2(\mathbb{Z}) \\ W &\in& \widehat{\mathbb{Z}^2} \\ {\left\vert [n] \right\rangle} &\in& \mathscr{H}_1 \mathrlap{\,.} \end{array} \right.$$

In other words, (14) enhances $\mathscr{H}_1$ to a representation of the semidirect product group $\widehat{\mathbb{Z}^2} \rtimes \mathrm{SL}_2(\mathbb{Z})$ with operation

$$(W,m)\cdot (W',m') \;\coloneqq\; \big( W \cdot m(W'),\, m \cdot m' \big) \,.$$

This representation (14) is just the modular action known from abelian Chern-Simons theory at even level $k$ (cf. Wen 1990 (5.3), Manoliu 1998a p 67, Gannon 2005 (3.1b)):

(16)$$\begin{array}{ccr} S\Big( {\big\vert [n] \big\rangle} \Big) &=& \frac{1}{\sqrt{\vert k \vert}} \sum_{ [\widehat n] } \, e^{ 2 \pi \mathrm{i} \tfrac{ \widehat{n} \, n }{ k } } {\big\vert [\widehat{n}] \big\rangle} \\ T\Big( {\big\vert [n] \big\rangle} \Big) &=& \underset{ \eqqcolon 1/c_k }{ \underbrace{ e^{ - \pi \mathrm{i}/12 } } } \, e^{ \tfrac{\pi \mathrm{i}}{k} n^2 } {\big\vert [ n ] \big\rangle} \end{array}$$

Proof. To see that (16) is indeed a representation of the modular group, we need to check that the relations (13) are satisfied:

First we find

$$\begin{array}{rcl} S^2 \Big({\big\vert [n] \big\rangle}\Big) &=& S \Big( \frac{1}{\sqrt{\vert k \vert}} \sum_{ [\widehat n] } \, e^{ 2 \pi \mathrm{i} \tfrac{ \widehat{n} \, n }{ k } } {\big\vert[\widehat{n}]\big\rangle} \Big) \\ &=& \sum_{ [\widehat{\widehat n}] } \, \underset{ \delta_0\big( [ n + \widehat{\widehat{n}} ] \big) }{ \underbrace{ \tfrac{1}{k} \sum_{ [\widehat n] } \, e^{ 2 \pi \mathrm{i} \tfrac{ \widehat{n} \, (n + \widehat{\widehat{n}}) }{ k } } } } \, {\big\vert[\widehat{\widehat{n}}]\big\rangle} \\ &=& {\big\vert[-n]\big\rangle} \end{array}$$

(where under the brace we evaluated the sum of roots of unity), which immediately implies the relation $S^4 = \mathrm{id}$ and thereby, with

$$T \circ S \Big( {\big\vert [n] \big\rangle} \Big) \;=\; \tfrac{1}{\sqrt{\vert k \vert} c_k} \displaystyle{ \sum_{ [\widehat n] } } \, e^{ \tfrac{ \pi \mathrm{i} }{k} ( \widehat{n}^2 + 2\widehat{n} \, n ) } {\big\vert[\widehat{n}]\big\rangle} \,,$$

also the relation $S^2 (T S) = (T S) S^2$. It just remains to show that $(T S)^3 = \mathrm{e}$ or equivalently that $S T S \,=\, T^{-1} S^{-1} T^{-1}$. Direct computation yields:

$$\begin{array}{l} T^{-1} S^{-1} T^{-1} {\vert n \rangle} \;=\; T^{-1} S^{-1} e^{ - \tfrac{\pi \mathrm{i}}{k} n^2 } {\vert n \rangle} \\ \;=\; T^{-1} \tfrac{1}{\sqrt{k}} \sum_{\widehat{n}} e^{ \tfrac{\pi \mathrm{i}}{k} ( - n^2 - 2 \widehat{n} n ) } {\vert \widehat{n} \rangle} \\ \;=\; \tfrac{1}{\sqrt{k}} \sum_{\widehat{n}} e^{ \tfrac{\pi \mathrm{i}}{k} ( - n^2 - 2 \widehat{n} n - \widehat{n}^2 ) } {\vert \widehat{n} \rangle} \\ \;=\; \tfrac{1}{\sqrt{k}} \sum_{\widehat{n}} e^{ - \tfrac{\pi \mathrm{i}}{k} (\widehat{n} + n)^2 } {\vert \widehat{n} \rangle} \end{array} \;\;\;\;\;\;\;\;\;\;\;\;\; \text{and} \;\;\;\;\;\;\;\;\;\;\;\;\; \begin{array}{l} S T S {\big\vert n \big\rangle} \;=\; S T \frac{1}{\sqrt{\vert k \vert}} \sum_{ \widehat n } \, e^{ 2 \pi \mathrm{i} \tfrac{ \widehat{n} \, n }{ k } } {\big\vert \widehat{n} \big\rangle} \\ \;=\; S \tfrac{1}{\sqrt{k}} \sum_{ \widehat n } \, e^{ \tfrac{\pi \mathrm{i}}{k} ( 2 \widehat{n} \, n + \widehat{n}^2 ) } {\big\vert \widehat{n} \big\rangle} \\ \;=\; \frac{1}{k} \sum_{ \widehat n ,\, \widehat{\widehat{n}} } \, e^{ \tfrac{\pi \mathrm{i}}{k} ( 2 \widehat{n} \, n + \widehat{n}^2 + 2 \widehat{\widehat{n}} \widehat{n} ) } {\big\vert \widehat{\widehat{n}} \big\rangle} \\ \;=\; \frac{1}{\sqrt{k}} \sum_{ \widehat{\widehat{n}} } \underbrace{ \tfrac{1}{\sqrt{k}} \sum_{ \widehat n } \, e^{ \tfrac{\pi \mathrm{i}}{k} (\widehat{n} + (n + \widehat{\widehat{n}}))^2 } } e^{ - \tfrac{\pi \mathrm{i}}{k} (n + \widehat{\widehat{n}})^2 } {\big\vert \widehat{\widehat{n}} \big\rangle} \,, \end{array}$$

where the term over the brace is in fact constant in $n$ and $\widehat{\widehat{n}}$ by the assumption that $k$ is even, because this implies that the summands are $k$-periodic:

(17)$$e^{ \tfrac{\pi \mathrm{i}}{k} (n + k)^2 } \;=\; e^{ \tfrac{\pi \mathrm{i}}{k} (n^2 + 2 n k + k^2) } \;=\; e^{ \tfrac{\pi \mathrm{i}}{k} n^2 } \underset{ 1\;if\;k\;even }{ \underbrace{ e^{ \pi \mathrm{i}(2n + k) } } } \;=\; e^{ \tfrac{\pi \mathrm{i}}{k} n^2 } \,.$$

This means that the last relation holds if the normalization factor $c_k$ is indeed fixed, as shown in (16), to this quadratic Gauss sum, which evaluated to (see there)

(18)$$c_k \;=\; \Big( \tfrac{1}{\sqrt{k}} \textstyle{ \sum_{ \widehat n } } \, e^{ \tfrac{ \pi \mathrm{i} }{k} \widehat{n}^2 } \Big)^{1/3} \;=\; \big(\sqrt{\mathrm{i}}\big)^{1/3} \;=\; e^{ \pi \mathrm{i}/12 } \,.$$

Finally to see that also the semidirect product of these two groups is represented in that (15) holds:

We may explicitly check this for $m \in \{S,T\}$ any one of the modular generators (12) by unwinding the above definitions:

$$\begin{array}{ccl} S(W_a) \cdot S\Big( {\big\vert [n] \big\rangle} \Big) &\equiv& W_b^{-1} \frac{1}{\sqrt{\vert k \vert}} \sum_{ [\widehat n] } \, e^{ 2 \pi \mathrm{i} \tfrac{ \widehat{n} \, n }{ k } } {\big\vert [\widehat{n}] \big\rangle} \\ &=& \frac{1}{\sqrt{\vert k \vert}} \sum_{ [\widehat n] } \, e^{ 2 \pi \mathrm{i} \tfrac{ (\widehat{n} + 1) \, n }{ k } } {\big\vert [\widehat{n}] \big\rangle} \\ &=& e^{ 2 \pi \mathrm{i} \tfrac{n}{k} } \, S\Big( {\big\vert [n] \big\rangle} \Big) \\ &=& S\Big( W_a {\big\vert [n] \big\rangle} \Big) \mathrlap{\,,} \end{array}$$

$$\begin{array}{ccl} S(W_b) \cdot S\Big( {\big\vert [n] \big\rangle} \Big) &\equiv& W_a \frac{1}{\sqrt{\vert k \vert}} \sum_{ [\widehat n] } \, e^{ 2 \pi \mathrm{i} \tfrac{ \widehat{n} \, n }{ k } } {\big\vert [\widehat{n}] \big\rangle} \\ &=& \frac{1}{\sqrt{\vert k \vert}} \sum_{ [\widehat n] } \, e^{2 \pi \mathrm{i} \tfrac{\widehat{n}}{k}} e^{ 2 \pi \mathrm{i} \tfrac{ \widehat{n} \, n }{ k } } {\big\vert [\widehat{n}] \big\rangle} \\ &=& S\Big( W_b {\big| [n] \big\rangle} \Big) \mathrlap{\,,} \end{array}$$

$$\begin{array}{ccl} T(W_a) \cdot T\Big( {\big\vert [n] \big\rangle} \Big) &\equiv& W_a \, e^{ \mathrm{i} \pi \tfrac{n^2}{k} } {\big\vert [n] \big\rangle} \\ &=& e^{2 \pi \mathrm{i} \tfrac{n}{k}} e^{ \mathrm{i} \pi \tfrac{n^2}{k} } {\big\vert [n] \big\rangle} \\ &=& T\Big( W_a \, {\big\vert [n] \big\rangle} \Big) \mathrlap{\,,} \end{array}$$

and finally, using (8):

$$\begin{array}{ccl} T(W_b)\cdot T\Big( {\big\vert [n] \big\rangle} \Big) &\equiv& W_b \, W_a \, e^{\pi \mathrm{i} \tfrac{1}{k}} \, e^{\pi \mathrm{i} \tfrac{n^2}{k}} {\big\vert [n] \big\rangle} \\ &=& e^{\pi \mathrm{i} \tfrac{ n^2 + 2n + 1 }{k}} {\big\vert [n + 1] \big\rangle} \\ &=& e^{\pi \mathrm{i} \tfrac{ (n+1)^2 }{k}} {\big\vert [n + 1] \big\rangle} \\ &=& T\Big( W_b {\big\vert [n] \big\rangle} \Big) \mathrlap{\,.} \end{array}$$

This completes the proof.  ▮

4. Related entries

• Heisenberg group

• 2-Cohomotopy moduli of surfaces

5. Literature

In group theory

Generally on the integer/discrete Heisenberg group in the algebra and group theory literature

• Soo Teck Lee, Judith A. Packer: The Cohomology of the Integer Heisenberg Groups, Journal of Algebra 184 1 (1996) 230-250 [doi:10.1006/jabr.1996.0258]

  (concerning its group cohomology)

• David S. Dummit, Richard M. Foote Ex. 11 on p. 35: Abstract Algebra, Wiley (2003) [ISBN:978-0-471-43334-7, pdf]

• Roman Budylin: Conjugacy classes in discrete Heisenberg groups, Sbornik: Mathematics 205 8 (2014) 1069–1079 [arXiv:1405.5499, doi:10.1070/SM2014v205n08ABEH004410]

• Daniel Bump, Persi Diaconis, Angela Hicks, Laurent Miclo, Harold Widom: An Exercise (?) in Fourier Analysis on the Heisenberg Group, Annales de la Faculté des sciences de Toulouse: Mathématiques, Serie 6, Volume 26 (2017) no. 2, pp. 263-288 [arXiv:1502.04160, numdam:AFST_2017_6_26_2_263_0]

• Cornelia Druţu, Michael Kapovich (appendix by Bogdan Nica): Geometric group theory, Colloquium Publications 63, AMS (2018) [ISBN:978-1-4704-1104-6, pdf]

• Jayadev S. Athreya, Ioannis Konstantoulas: Lattice deformations in the Heisenberg group, Groups, Geometry and Dynamics 14 3 (2020) 1007–1022 [arXiv:1510.01433, doi:10.4171/ggd/572]

• Uri Bader, Vladimir Finkelshtein, §5 of: On the horofunction boundary of discrete Heisenberg group, Geom Dedicata 208 (2020) 113–127 [arXiv:1904.11234, doi:10.1007/s10711-020-00513-x]

• Ruiwen Dong, p 9 of: Recent advances in algorithmic problems for semigroups, ACM SIGLOG News 10 4 (2023) 3-23 [arXiv:2309.10943, doi:10.1145/3636362.3636365]

On (invertibility in) the group algebra:

• Martin Göll, Klaus Schmidt, Evgeny Verbitskiy: A Wiener Lemma for the discrete Heisenberg group: Invertibility criteria and applications to algebraic dynamics, Monatsh Math 180 (2016) 485–525 [doi:10.1007/s00605-016-0894-0, arXiv:1603.08225]

On the representation theory with an eye towards quantum information theory:

• E. Floratos, I. Tsohantjis: Complete set of unitary irreps of Discrete Heisenberg Group $H W_{2^s}$ [arXiv:2210.04263]

On the automorphism group:

• D.V. Osipov: Discrete Heisenberg group and its automorphism group, Mathematical Notes 98 1 (2015) 185-188 [arXiv:1505.00348, doi:10.1134/S0001434615070160]

See also:

• Yves Benoist: Positive Harmonic Functions on the Heisenberg group I, in: Proceedings of European Congress of Mathematics 2021, EMS Press (2023) 181-198 [arXiv:1907.05041, doi:10.4171/8ecm/17]

• Yves Benoist: Positive harmonic functions on the Heisenberg group II, Journal de l’École polytechnique - Mathématiques 8 (2021) 973-1003 [doi:10.5802/jep.163, numdam:10.5802/jep.163, hal:02400005]

In modular/Chern-Simons theory

In relation to $U(1)$-current algebra (WZW-model):

• Ludwig D. Fadeev: Discrete Heisenberg-Weyl Group and modular group, Lett Math Phys 34 (1995) 249–254 [doi:10.1007/BF01872779]

As describing the phase space of abelian Chern-Simons theory on closed Riemann surfaces (and its relation to skein relations and theta functions):

• Răzvan Gelca, Alejandro Uribe: From classical theta functions to topological quantum field theory, in: The Influence of Solomon Lefschetz in Geometry and Topology: 50 Years of Mathematics at CINVESTAV, Contemporary Mathematics 621, AMS (2014) 35-68 [arXiv:1006.3252, doi;10.1090/conm/621, ams:conm-621, slides pdf, pdf]

• Răzvan Gelca, Alastair Hamilton: Classical theta functions from a quantum group perspective, New York J. Math. 21 (2015) 93–127 [arXiv:1209.1135, nyjm:j/2015/21-4]

• Răzvan Gelca, Alastair Hamilton: The topological quantum field theory of Riemann’s theta functions, Journal of Geometry and Physics 98 (2015) 242-261 [doi:10.1016/j.geomphys.2015.08.008, arXiv:1406.4269]

On group actions of the mapping class group of closed oriented surfaces on integer Heisenberg groups:

• Awais Shaukat, Christian Blanchet: Weakly framed surface configurations, Heisenberg homology and Mapping Class Group action, Arch. Math. 120 (2023) 99–109 [arXiv:2206.11475, doi:10.1007/s00013-022-01793-3]

• Christian Blanchet, Martin Palmer, Awais Shaukat: Action of subgroups of the mapping class group on Heisenberg homologies, Contemporary Mathematics [arXiv:2306.08614]

• Martin Palmer: Representations of the Torelli group via the Heisenberg group, talk at Workshop for Young Researchers in Mathematics – 10th ed. (2021) [pdf, pdf]

• Christian Blanchet: Heisenberg homologies of surface configurations, talk at Geometric/Topological Quantum Field Theories Workshop @ CQTS NYUAD (2023) [slides: pdf, video:YT]

The above discussion of irreps and modular automorphisms related to abelian Chern-Simons theory follows:

• Hisham Sati, Urs Schreiber: Engineering of Anyons on M5-Probes via Flux-Quantization, Lecture Notes (2025)